Equation (REF ) arises from several canonical hydrodynamic models.
One typical model is the viscous and resistive MHD system
(i.e., \(\alpha _1=\alpha _2=1\) ),
which is a classical macroscopic model to describe the motion of conductive fluid in an electromagnetic field
(cf. [1]}, [2]}, [3]}, [4]}).
In the case where \(B\equiv 0\) , (REF ) reduces to the hyperviscous Navier-Stokes equations
\( \left\lbrace \aligned &\partial _tu+\nu (-\Delta )^{\alpha } u+(u\cdot \nabla )u + \nabla P=0, \\& {\mathrm {div}}u = 0.\endaligned \right.\) 
